Pseudolite-aided method for precision kinematic positioning

ABSTRACT

A technique using pseudo-satellites, or &#34;pseudolites,&#34; for resolving whole-cycle ambiguity that is inherent in phase-angle measurments of signals received from multiple satellite-based transmitters in a global positioning system. The relative position of a secondary receiving antenna with respect to a reference antenna is approximately known or approximately initially determined and then measurements from a minimum number of satellites are used to determine an initial set of potential solutions to the relative position of the secondary antenna that fall within a region of uncertainty surrounding the approximate position. Redundant measurements are taken from one or more pseudolites, and used to progressively reduce the number of potential solutions to close to one. Each pseudolite is positioned such that its angular position as seen from the secondary antenna changes rapidly as the antenna is moved along an intended path, thus providing rapidly changing redundant measurements, which are used to eliminate false solutions more quickly. The disclosed technique produces rapid elimination of false solutions and permits real-time computation of relative position in kinetic positioning applications.

CROSS-REFERENCE TO RELATED APPLICATION

This is a continuation-in-part of application Ser. No. 07/558,911 filedJul. 27, 1990, Ronald R. Hatch, entitled "Method and Apparatus forPrecision Attitude Determination," now issued as U.S. Pat. No. 5,072,227which was a divisional of application Ser. No. 07/413,411, filed Sep.26. 1989 entitled "Method and Apparatus for Precision KinematicPositioning," now issued as U.S. Pat. No. 4,963,889.

BACKGROUND OF THE INVENTION

This invention relates generally to positioning systems using signalsbroadcast from a plurality of orbiting satellites and, moreparticularly, to satellite-based differential positioning systems thatdetermine the position coordinates of one receiver, referred to as aremote receiver, relative to the known position of another, referred toas the reference receiver.

There are two types of applications in which measurements of this kindmay be usefully employed. One is referred to as kinematic positioning,in which the remote receiver may be moved with respect to the referencereceiver, and the distance between the remote receiver and referencereceiver is, therefore, not initially known. The other application isattitude determination, in which the distance between the remote andreference receivers is fixed, and the position of the remote receiver isused to determine the angular position of a line or chord joining thetwo receiver antennas. If three antennas are used instead of two, theangular position of a plane intersecting the three antennas can bedetermined from the relative positions of two of the antennas withrespect to the third, used as a reference. Attitude determination hasapplication in navigation systems on or above the earth. Using aplatform having three antennas, the roll, pitch and yaw angles of aplatform supporting the three antennas can be determined.

Satellite-based positioning systems, such as the Global PositioningSystem (GPS), provide a now widely used means for accurately determiningthe position of a receiver in three-dimensional space. These systemshave numerous practical applications and, depending on the time durationover which measurements are taken, they can determine a receiver'sposition to subcentimeter accuracy.

In GPS, a number of satellites orbiting the earth in well-defined polarorbits continually broadcast signals indicating their precise orbitalpositions. Each satellite broadcasts two modulated carrier signals,designated L₁ and L₂. The same two frequencies are used in transmittingfrom all of the satellites, but the satellites have unique pseudorandomdigital codes that are used to modulate the L₁ and L₂ carriers. Eachsatellite signal is based on a precision internal clock, and themultiple clocks are effectively synchronized by ground-based stationsthat are a necessary part of GPS The receivers detect superimposedmodulated L₁ and L₂ signals and measure either or both of the code andcarrier phase of each detected signal, relative to their own internalclocks. Even though a receiver clock is not synchronized with thesatellite clocks, a receiver can nevertheless determine the"pseudorange" to each satellite based on the relative time of arrival ofthe signals, and the receiver position can then be mathematicallydetermined from the pseudoranges and the known positions of thesatellites. The clock error between the receiver's time reference andthe satellite clocks can be eliminated by the availability of signalsfrom an additional satellite. Thus, to solve for three unknownpositional coordinates and the clock error requires the acquisition offour satellite signals.

GPS satellites provide two types of signals that can be used forpositioning. The pseudorandom digital codes, referred to as the C/A codeand the P code, provide unambiguous range measurements to eachsatellite, but they each have a relatively long "wavelength," of about300 meters and 30 meters, respectively. Consequently, use of the C/Acode and the P code yield position data only at a relatively coarselevel of resolution. The other type of signal that can be used forposition determination is the carrier signals themselves. The L₁ and L₂carrier signals have wavelengths of about 19 and 24 centimeters,respectively. In a known technique of range measurement, the phase ofone of the carrier signals is detected, permitting range measurement toan accuracy of less than a centimeter. The principal difficulty withusing carrier signals for range measurement is that there is an inherentambiguity that arises because each cycle of the carrier signal looksexactly alike. Therefore, the range measurement has an ambiguityequivalent to an integral number of carrier signal wavelengths. Varioustechniques are used to resolve the ambiguity. In a sense, the presentinvention is concerned with a novel technique for this type of ambiguityresolution.

In absolute positioning systems, i.e. systems that determine areceiver's position coordinates without reference to a nearby referencereceiver, the process of position determination is subject to errorsfrom a number of sources. These include orbital and ionospheric andtropospheric refraction errors. For attitude determination applications,the receivers are located so close together that these errors arecompletely negligible, that is to say they affect both or all threereceivers substantially equally. For greater receiver spacings, as inkinematic positioning applications, such errors become significant andmust be eliminated. It will be appreciated that the problems of attitudedetermination and kinematic positioning are closely analogous. Thesignificant difference is that, in attitude determination, the distancebetween receivers is constrained. As a result, the receivers can beoperated from a single reference clock. In a general sense, however, theattitude determination application is simply a more restricted form ofthe kinematic positioning problem.

In many kinematic positioning applications, a reference receiver locatedat a reference site having known coordinates is available for receivingthe satellite signals simultaneously with the receipt of signals by theremote receiver. Depending on the separation distance, many of theerrors mentioned above will be of about the same magnitude and willaffect the various satellite signals they receive substantially equallyfor the two receivers. In this circumstance, the signals receivedsimultaneously by two receivers can be suitably combined tosubstantially eliminate the error-producing effects of the ionosphere,and thus provide an accurate determination of the remote receiver'scoordinates relative to the reference receiver's coordinates.

To properly combine the signals received simultaneously by the referencereceiver and the remote receiver, and thereby eliminate theerror-producing effects, it is necessary to provide an accurate initialestimate of the remote receiver's coordinates relative to the referencereceiver. By far the easiest way to obtain the initial relative positionof the remote receiver is to locate it at a pre-surveyed marker.Unfortunately, such markers are seldom available in many practicalapplications.

Another method often used to obtain an accurate initial relativeposition is to exchange the receivers and antennas between the referenceand remote sites while both continue to operate. This results in anapparent movement between the two antennas of twice the vectordifference between them. This apparent movement can be halved and usedas the initial offset between them. The approach works well as long asthe remote receiver is in the immediate vicinity of the reference site.Unfortunately, any time the satellite signals are lost the initialposition must be reestablished, which means that the remote receivermust be returned to the control site or to a nearby marker. This isimpractical in many applications, such as photogrammetric survey byaircraft.

In a prior patent to Ronald R. Hatch, U.S. Pat. No. 4,812,991, a methodusing carrier smoothed code measurements to determine an increasinglyaccurate initial position was described. This technique had theadvantage that it did not require the remote receiver to remainstationary while the initial relative position was established. Thedisadvantage of that method is twofold. First, it is not aninstantaneous method of establishing the initial position and can takeseveral minutes of data collection and processing to accomplish thetask. Second it requires access to the the precise (P) code modulationon the L₂ carrier frequency. Unfortunately, the United States Departmentof Defense has reserved the right to limit access to P code modulationby encrypting the P code prior to transmission from each satellite.Therefore, the method described in the prior patent cannot be used whenaccess to the P code is denied.

The aforementioned U.S. Pat. No. 4,963,889 to Hatch describes and claimsa satisfactory solution to the problems discussed above, whereinmeasurements from a minimum number of satellites are used to determinean initial set of potential solutions to the position of the secondaryantenna. Redundant measurements from additional satellites are then usedto progressively reduce the number of potential solutions to close toone. The only difficulty with the proposed solution occurs when thesolution is required very rapidly, and there are not enough satellitesin the field of view of the antenna to achieve a rapid solution. Forexample, one application of kinematic position determination is ininstrument landing systems for aircraft. Ideally, the position of theaircraft must be determined in "real time," and waiting for satellitesto move to a new angular position, to provide additional measurementsthat reduce the uncertainty of the position estimate, is not a practicaloption. Accordingly, there is still need for improvement in this field,to provide a technique that reduces the time needed to eliminate falseposition solutions. The present invention fulfills this need.

SUMMARY OF THE INVENTION

The present invention resides in a method for determining thecoordinates of a remote receiver antenna relative to a referencereceiver antenna, using signal received from one or more pseudosatellites, or pseudolites. The remote or secondary antenna is freelymovable with respect to the reference antenna, and the process ofdetermining its position is generally referred to as the kineticpositioning problem.

Briefly, and in general terms, the method of the invention includes thesteps of making carrier phase measurements based on the reception of acarrier signal from each of a plurality N of satellites, where N is theminimum number of satellites needed to compute the relative position ofthe secondary antenna, and deriving from the carrier phase measurementsan initial set of potential solutions for the relative position, whereinthe initial set of potential solutions all fall within athree-dimensional region of uncertainty, and wherein multiple potentialsolutions arise because of whole-cycle ambiguity of the carrier signal.The method further includes positioning at least one pseudolitetransmitter on the ground at a location near an intended path of travelof the secondary antenna, and making redundant carrier phasemeasurements based on the reception of a carrier signal from the atleast one pseudolite. The final step of the method is eliminating falsesolutions from the initial set of potential solutions, based on acomparison of the redundant carrier phase measurements with the initialset of potential solutions, to reduce number of potential solutions toclose to one, whereby the number of potential solutions is not reducedby use of the redundant carrier phase measurements.

The method of the invention may further include making redundant carrierphase measurements based on the reception of a carrier signal from otheradditional satellites, and eliminating other false solutions from theset of potential solutions, based on a comparison of the additionalredundant carrier phase measurements with the initial set of potentialsolutions. The method may also include comparing items in the set ofpotential solutions, including those obtained from the redundant phasemeasurements, with solutions obtained in a prior time interval, toprovide another basis for eliminating false solutions. The location ofthe at least one pseudolite provides for a rapidly changing angulargeometry and reduces the time needed to eliminate false solutions.

More specifically, the step of deriving an initial set of potentialsolutions includes locating points of intersection of planar carrierwavefronts defining possible locations of the secondary antenna withinthe region of uncertainty, and the step of eliminating false solutionsincludes locating a set of planar carrier wavefronts from the at leastone pseudolite such that the wavefronts also define possible locationsof the secondary antenna within the region of uncertainty, selecting,for each of the initial set of potential solutions, a planar carrierwavefront from the additional satellite such that the selected wavefrontis the one closest to the potential solution, and disregarding eachpotential solution for which the closest wavefront from the additionalsatellite is spaced from the potential solution by more than a selectedthreshold.

The initial set of potential solutions is initially stored in a localtangent coordinate system, x, y, z, where z is a vertical axis. Themethod may further comprise the additional step of rotating thecoordinate system of the set of potential solutions, to point the z axistoward the additional satellite and thereby facilitate the step ofeliminating false solutions. For kinematic positioning, the secondaryantenna is freely movable with respect to the reference antenna, and themethod further comprises the initial step of determining the approximateinitial relative position of the secondary antenna.

In accordance with another aspect of the invention, a pseudolite signalmay also provide a communication link between a reference receiver andthe remote receiver. The reference receiver or receivers may be locatedclose to the pseudolite transmitter and the pseudolite signal can bemodulated in the same way as satellite signals to provide thecommunication link from the reference receiver to the remote receiver.For real time operation such a communication link is needed to provideto the remote receiver raw measurements or correction data generated atthe reference receiver.

It will be appreciated from this summary that the present inventionrepresents a significant advance in the field of satellite-basedpositioning systems. In particular, the invention resolves whole-cycleambiguity inherent in carrier phase measurements from satellite signals,and does so in such an efficient and rapid manner that the relativeposition of an antenna can be computed on a real-time basis to a highdegree of accuracy. The most significant aspect of the invention is thatit uses redundant measurements from one or more pseudolites to reducethe number of potential solutions to the relative positioning problem,rather than to increase the number of potential solutions. Other aspectsand advantages of the invention will become apparent from the followingmore detailed description, taken in conjunction with the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram depicting carrier phase ambiguities in signalsreceived from a single satellite by two antennas of fixed spacing;

FIG. 2 is a diagram similar to FIG. 1 but also showing the phaseambiguities of signals received from a second satellite in the sameplane as the first;

FIG. 3 is an elevational view showing the phase ambiguities for the caseof signals received from two orthogonally positioned satellites, by areference antenna and a secondary antenna movable in three-dimensionalspace at a fixed distance from the reference antenna;

FIG. 4 is a view similar to FIG. 3, but showing phase ambiguities forthe case of two non-orthogonal satellites;

FIG. 5 is a view similar to FIG. 3, but showing the effect of phaseambiguities for the case of three satellites;

FIG. 6A is a block diagram of apparatus used for attitude determination;

FIG. 6B is a block diagram of apparatus used in accordance with theinvention for kinetic position determination; and

FIG. 7 is a flowchart of the process steps performed in accordance withthe invention.

DESCRIPTION OF THE PREFERRED EMBODIMENT Overview

As shown in the drawings for purposes of illustration, the presentinvention is concerned with a technique for determining from receivedGlobal Positioning System (GPS) signals the position of a remotereceiver antenna relative to a reference receiver antenna. The relativeposition information can be used in a kinematic positioning system or inan attitude determination system.

As discussed in the background section of this specification, attitudedetermination can be logically viewed as a special case of kinematicpositioning, wherein the distance between the reference and remoteantennas is fixed and known. For this reason, explanation of theinvention is facilitated by first describing its application to theattitude determination problem.

FIG. 1 is a diagrammatic depiction of two antennas, referred to as A andB, and the attitude determination problem is to find the angularelevation of the line AB between the antennas, by measuring the phasedifference between carrier signals arriving at the antennas. Horizontallines, indicated by reference numeral 10, represent wavefronts of acarrier signal received from a satellite located immediately overhead.It is assumed that the antennas are separated by a known distance equalto five carrier wavelengths, or approximately one meter at the GPS L₁carrier frequency. It is also assumed that the phase difference measuredbetween the two antennas is zero. Because the carrier phase repeatsidentically every wavelength, there are twenty locations, of whicheleven are shown at B1 through B11, where antenna B will see the samephase as antenna A. If the distance AB were less than a carrierwavelength, there would be no ambiguity, but the pointing accuracy wouldbe five times worse. If the length AB were greater than fivewavelengths, the accuracy would improve but the number of ambiguitieswould increase.

One method for resolving the ambiguity is to move one antenna from A toB and to measure the total phase change. For example, in moving from Ato B6 there would be no phase change, since B6 is located on the samewavefront as A. Similarly, in moving from A to B5 there is a phasechange of 360 degrees. By observing the total phase change in movingfrom A to the actual position of B, including the number of wholecycles, any phase ambiguity is immediately resolved. Unfortunately, thisapproach is impractical for many applications.

Another approach, which is the one used in this invention, is to resolvethe ambiguity by utilizing signals from a second satellite. FIG. 2 showshow this approach is applied in the simple two-dimensional attitudedetermination problem depicted in FIG. 1. The measured phase differencefor signals from a second satellite may not necessarily be zero, andsome other value is assumed here, as might be deduced from the fact thatnone of the wavefronts 12 from the second satellite passes through thefixed antenna A. The wavefronts 12 are drawn such that antenna A isdisplaced from one of the lines 12, which represent zero phase, by anamount corresponding to the measured phase difference between antennas Aand B, for signals from the second satellite. Thus, any of the lines 12that passes through one of the points B1-B11 represents a possiblesolution. As will be observed, the only exact solution is at B6. For thepoints B2, B4 and B10, there is a poor fit between the point and thelines 12, and none of the other B points fits as precisely to a line 12as does B6.

If the line AB may be positioned at any angle in three-dimensionalspace, the position of the remote antenna B is defined by the surface ofa sphere with a radius equal to the line AB. This is shown intwo-dimensional form in FIG. 3, for the case in which AB is one meter inlength and the carrier is L₁, with a wavelength of 19 centimeters. Thecircle 14 in FIG. 3 is an elevational view of a sphere of one-meterradius. Horizontal lines 16 represent wavefronts of the L₁ carriersignal from an overhead satellite, and vertical lines 18 representwavefronts of the L₁ carrier signal from a second satellite located atthe horizon. The lines 16 and 18 are actually planes of equal phase,which intersect the surface of sphere. These planes intersect each otheralong straight lines 20, indicated as points in the drawing. Each of thelines 20 intersects the spherical surface at two points, which are seenas being coincident in the drawing. For the example given, there are atotal of 11×11 or 121 lines 20, but only 94 of these fall within theprojection of the sphere seen in FIG. 3. Thus there are only 188 pointsof intersection between the lines 20 and the surface of the sphere. Eachsuch intersection represents a possible solution to the wavelengthambiguity of the carrier signal. This is to be distinguished, of coursefrom the possible resolution with which angular position can bedetermined. Phase differences can be detected to much smallerresolutions than one full wavelength.

FIG. 4 is similar to FIG. 3 except that it shows wavefronts from twonon-orthogonal satellites, with the wavefronts indicated by 16 and 18'.The principal difference is that there are fewer intersecting lines 20',giving a total of 126 points of intersection between the lines 20' andthe surface of the sphere.

FIG. 5 shows how the availability of signals from a third satellitefurther reduces the ambiguity inherent in FIG. 3. The points ofintersection of the lines 20 and the surface of the sphere are indicatedin FIG. 5 as small square areas 22. In practice, the points 20 cannot beachieved because of uncertainties of phase measurement. Hence therepresentation as small square regions 22. Wavefronts 24 emanate from athird satellite which is assumed, for ease of illustration, to be in thesame plane as the first and second satellites. Again it is assumed thatthere is a zero phase difference between the two antennas receiving thisthird satellite signal. Therefore, any nonintersection of the planes 24with the square regions 22 may be eliminated as a possible solution.This reduces the possible solutions to 28 points.

The same principle discussed above can also be applied to the kinematicpositioning application. The principal difference is that, instead ofbeing the surface of a sphere, the region of uncertainty is athree-dimensional volume. As already mentioned, four satellite signalsare needed to determine the three-dimensional relative position and theclock phase or clock error. However, the clock phase may be considered a"nuisance variable" for present purposes, since the clock phase does notneed to assume multiple wholecycle values. The number of potentialsolutions in an uncertainty volume of one cubic meter is then a maximumof 11³ or 1,331. When more than four satellites are available, redundantsatellites can be used to reduce the number of these potentialsolutions.

In general, kinematic position determination includes three major steps:approximately determining the relative position of a remote receiverwith respect to a reference receiver, determining all potentialsolutions that are present in an uncertainty region surrounding theapproximate solution, and eliminating false potential solutions in thepotential solutions. Specific techniques for performing the first twosteps in the kinematic positioning application will be discussed laterin this specification.

In accordance with the invention, the step of eliminating falsepotential solutions includes taking successive measurements using atleast one "pseudolite." A pseudolite is a fixed ground-basedtransmitter, which functions in a communications sense exactly like asatellite transmitter, except that it never moves from its position onthe ground.

If the position of the pseudolite is appropriately chosen, such at oneside of an aircraft runway, an approaching aircraft using the method ofthe invention will experience a rapid change in the sensed angularposition of the pseudolite. Consequently, false solutions can be morerapidly eliminated than if one had to wait for the angular positions ofthe satellites to change.

The step of determining all of the potential solutions to the relativeposition of the remote antenna involves carrier phase measurementsobtained at both antennas, using the minimum number of satellitesrequired to obtain a relative solution. These carrier phase measurementsare indicative of the range from the receiver antennas to thesatellites, but they are ambiguous at the whole cycle level. However, ifthe relative position of the remote antenna with respect to thereference antenna were known precisely, then the difference in range tothe satellite from each of the antennas could also be preciselydetermined, and from this difference in range the number of whole cyclesof phase in the difference of the carrier phase measurements could bycomputed. But, since the relative position of the remote antenna isknown only approximately, the range difference between the two antennasand each satellite is also ambiguous at the whole cycle level. That isto say, several different values of the whole cycle difference in rangecould possibly be the true whole cycle value of the difference.

As discussed above with reference to FIG. 1, a set of potentialsolutions can be found which are close enough to the approximaterelative position to have a reasonable probability of being the truerelative position. In fact, any of the combinations of whole cyclevalues of carrier phase difference which result in potential solutionswithin an uncertainty region surrounding the approximate solution couldbe the true solution.

The number of potential solutions is a function of the size of theuncertainty region and of the wavelength of the carrier signal. The L₁carrier has a wavelength close to 19 centimeters, the L₂ carrier has awavelength close to 24.4 centimeters, and the wavelength of thedifference frequency L₁ -L₂ has a wavelength of about 86 centimeters.Therefore, in any given uncertainty region there are fewer potentialsolutions if one can use the difference of the carrier phase at the twofrequencies than if one must use the L₁ or the L₂ carrier phasemeasurements alone. This means that two-frequency receivers arepreferred to single-frequency receivers for application of theinvention.

In kinematic positioning, the minimum number of satellites needed tosolve for the relative position of the remote site with respect to thereference site is four. Measurements to four satellites are requiredbecause there are three dimensions of relative position and anadditional unknown in the relative clock phase of the two receivers.Consider the difference in carrier phase measurements to one of the foursatellites obtained at the two receivers. This difference in the carrierphase measurements across the two sites will be referred to as the"first-difference" of the two measurements. If, for example, theapproximate relative position of the two receivers is known sufficientlyaccurately to be sure that it is within plus or minus 4.3 meters of thetrue solution, and assuming two-frequency receivers are employed (whichallow the use of the 86-centimeter wavelength for the differencefrequency), then a total of ten different whole-cycle values could beadded to the first-difference of the carrier phase measurements andstill agree with the approximate relative position within plus or minus4.3 meters. In relative to FIGS. 1-4, the region of uncertainty has aradius of 4.3 meters, and ten wavefronts of wavelength 86 centimeterswill fall within the region of uncertainty.

Continuing with this example, if ten different whole-cycle values can beadded to the first-differences from each of the four satellites, thenone might expect 10⁴ or 10,000 possible solutions within the uncertaintyregion. Indeed, some authors have defined the number of possibleacceptable solutions in exactly this fashion. However, only 1,000 of thepotential solutions in this example are independent position solutions.The other 9,000 solutions differ from one of the independent solutionsonly in the whole-cycle value of the clock phase of the two receivers.But this relative whole-cycle value of the clock phase is of nopractical interest, and it can be eliminated from consideration in thekinematic positioning application, to prevent any effect on the solutionfor the relative position coordinates of interest.

Removal of the whole-cycle clock phase measurements in the kinematicpositioning application is best illustrated by forming what will bereferred to as second-difference measurements. If one of the satellitesis chosen as a reference satellite, and the first-differencemeasurements for this satellite are subtracted from the first-differencemeasurements for the other three satellites, there remain threesecond-difference measurements, which are independent of the relativeclock phase of the two receivers. Using the same example of uncertaintyin approximate position as was used above, there are ten acceptablewhole-cycle values that can be added to each of the threesecond-difference measurements. This leads directly to a conclusion thatthere are 10³ or 1,000 potential solutions in the vicinity of theapproximate relative position.

The situation is simpler in the attitude determination application, inwhich the relative separation distance is generally so short that thetwo receivers can be designed such that the same clock drives both ofthe receivers. Therefore, there is no unknown relative clock phase thatneed to be made part of the solution. This means that thesecond-difference of the measurements as described above is not requiredin the attitude determination application. In addition, the separationdistance of the two antennas is generally known and can be used toconstrain the relative position of the two antennas and leave only atwo-dimensional uncertainty region, corresponding to the surface of asphere, as discussed with reference to FIGS. 3 and 4. Thus, only twosatellites are needed to define all the potential solutions that canexist on the two-dimensional uncertainty region.

In brief, then, the kinematic positioning application reduces to athree-dimensional uncertainty problem, requiring three satellites forposition measurement and one to handle clock uncertainty, to define allof the potential solutions. The attitude detection application is atwo-dimensional uncertainty problem, requiring two satellites to defineall of the potential solutions.

The final broad step in the method of the invention is eliminating falsepotential solutions based on redundant information from additionalsatellites and, in the case of the present invention, one or morepsuedolites. Each additional redundant measurement, whether it is asecond-difference measurement in a kinematic positioning application ora first-difference measurement in an attitude determination application,could have multiple whole-cycle values added to it which would stillcause the surface defined by the measurement to pass through theuncertainty region. If each of these possible whole-cycle values wereused to modify each of the potential solutions, then the number ofpotential solutions would increase by a large factor. However, only oneof the whole-cycle values will result in a surface that passes closer aspecific potential solution than any other whole-cycle value. Thisclosest to whole-cycle value (from the additional satellite data) isselected for each of the potential solutions, and is used to modify theredundant measurement for inclusion in that same potential solution. Byusing only the closest whole-cycle value for the redundant measurement,the number of potential solutions is not increased.

However, even the closest whole-cycle value may still result in theredundant measurement disagreeing substantially with the potentialsolution. When the redundant measurements as modified by the appropriatewhole-cycle value is incorporated into the potential solution, it willresult in a large root mean square (rms) residual if it is insubstantial disagreement with the original potential solution. Since thepotential solution after the redundant measurement is overdetermined,the rms residual error is a quantitative measure of the amount ofdisagreement. A threshold can be selected which, when the rms residualerror exceeds it, results in the potential solution being declared afalse solution and removed from the set of potential solutions. Thethreshold can be adjusted as a function of the carrier phase measurementnoise, to result in a high probability of rejecting false solution whilemaintaining an acceptable low probability of rejecting a true solution.

If enough redundant measurements are available, all of the falsesolutions can be eliminated with one set of essentially instantaneousmeasurements from the multiple satellite signals, leaving only the truesolution. Even when the set of redundant measurements is insufficient toeliminate all of the false solutions instantaneously, and residual falsesolutions remain in the set of potential solutions, the redundantmeasurements are still of great benefit. As the relative geometry of thesatellite that provides the redundant measurements changes due to motionof either the satellites or motion of the remote receiver, the specificfalse solutions which are not eliminated from the set of potentialsolutions will change. Only the true solution will continue to be acommon subset of the set of potential solutions. This allows the truesolution to identified more quickly if the geometry changes quickly,even when the redundant measurements are insufficient to identify thetrue solution instantaneously. In the method of the invention, at leastone pseudolite transmitter is placed in a position that ensures a rapidchange in the angular position of the pseudolite, as viewed from amoving secondary antenna. For example, if the application of theinvention is an aircraft instrument landing system, a pseudolite may beconveniently positioned close to, but to one side of an aircraft runway.An aircraft on a final approach, will view the pseudolite at a rapidlychanging angular position. Consequently, false position solutions can beeliminated more rapidly than if only true satellite measurements wereused.

The mathematical relationships underlying the concepts described in thisoverview of the invention will be set out in the following descriptivesections. For convenience, the remainder of this detailed description isdivided into two sections, relating to attitude determination andkinematic positioning, respectively. As mentioned earlier, attitudedetermination may be viewed as a special case of kinematic positiondetermination. To facilitate explanation of the invention, attitudedetermination is described first. However, the present invention isconcerned only with an improvement relating to the kinematic positioningapplication.

Attitude Determination System

As shown in FIG. 6A, apparatus for accurately determining theorientation of one or more secondary antennas 30 with respect to areference antenna 32, uses signals broadcast from a plurality oforbiting satellites 34. The apparatus is particularly useful as part ofthe Global Positioning System (GPS), in which each satellite broadcaststwo separate carrier signals, denoted L₁ and L₂, each modulated by aseparate pseudorandom digital code. The entire structure supporting thesecondary and reference antennas 30, 32 and an associated receiver 36 isallowed to move, but the distance between the antenna is held fixed.Typical separation distances are on the order of one meter. For the foursatellites 34 depicted, there are eight separate links 38 formed betweenthe satellites and the antennas 30, 32.

The L₁ and L₂ carrier signals broadcast by the four satellites 34 are onthe same two frequencies, but each such carrier signal is modulated apseudorandom digital that is unique to the satellite from which it istransmitted. This provides one method by which the satellite signals maybe separated, even though received using omnidirectional antennas. Thetwo antennas 30, 32 are connected to separate receivers or, asillustrated, the same receiver 36, which is driven by a a singlereference clock (not shown). The receiver 36 separates the superimposedmultiple satellite signals from each other, and demodulates the signalsto obtain the pseudorandom digital code. The receiver 36 derives twotypes of measurements from the received signals. These are referred toas "code measurements" and "carrier measurements." The pseudorandomdigital code signals provide an unambiguous measure of the distance toeach satellite, referred to as the "pseudorange." This is notnecessarily the same as the actual range to the satellite, because oflack of time clock synchronism between the satellite and the receiver.However, the pseudoranges of the four satellites can be used toeliminate the clock error and derive the position of a receiver.However, the relatively long effective "wavelength" of the pseudorandomcodes limits the accuracy of the measurements, which is why measurementsof carrier phase are made by the receiver. Carrier phase measurementscan provide positional data to subcentimeter accuracy, provided there issome way to eliminate the inherent whole-cycle ambiguity of carrierphase measurements.

For operation of the present invention, the receiver 36 should becapable of receiving at least the L₁ carrier. As previously discussed,there are advantages in receiving the L₂ carrier as well, but this isnot an absolute requirement. The code and carrier phase measurements aretransmitted to a processor, indicated at 38, which may be incorporatedinto the receiver 36, where the attitude determination computations areperformed. Typically, the code and carrier measurements are made at arate of ten times per second. The processor 38 may process all of themeasurements in real time, or, if it has insufficient processing speed,the measurements may be sampled by selecting only every Nth measurement,where N is often on the order of 10, or the measurements may becompacted by fitting sequential groups of N measurements to a polynomialin time, and the polynomial value at its mid-point point in time is usedas a smoothed measurement.

On start-up or if signal reception from some of the satellites has beenlost, an initialization procedure must be implemented. This procedure iscomputation intensive, and ongoing measurements will be sampled orcompacted if processing power is limited.

The first computations to be performed in an attitude determinationapplication are to compute the entire set of potential solutions basedon the carrier phase measurements from two satellites. One of thesignificant features of the method described is the use of only two ofthe satellites to determine the set of potential solutions, rather thanmore than two satellites. The two satellites used for this computationusually are chosen as the two of highest elevation, or the two that areclosest together in angular separation as seen from the referenceantenna 30. These choices tend to minimize the total number of potentialsolutions that must be computed in the initial set. The two satelliteschosen are used to compute all possible sets of coordinates of thesecondary antenna 32 relative to the reference antenna 30, which satisfythe carrier phase measurements obtained from the two selectedsatellites, subject to the separation distance constraint.

The process of determining the set of all possible solutions wasdescribed in general terms with reference to FIGS. 3 and 4. The planes16 and 18 define different whole-cycle values of the first-differencemeasurements from the two satellites, and the intersections of the lines20 with the spherical surface comprising the region of uncertaintydefine potential solutions to the relative position of the secondaryantenna 30. It will be recalled that, for a wavelength of 19 centimeters(L₁) and a one-meter antenna spacing, the number of potential solutionsresulting from measurements from two satellites is 188. Although thereare 121 lines of intersection between the two sets of planes 16 and 18,only a maximum of 89 of these intersect the spherical surface definingthe region of uncertainty. Hence there are 188 potential solutions inthis example. Various mathematical techniques may be used to obtain thepotential solutions, and to eliminate false solutions. One of thesetechniques will now be described.

Some equivalent mathematical techniques to compute the potentialsolutions include the use of a Kalman or Magil filter, the use ofsequential least squares, and the use of Square Root Information Filters(SRIF). Information on the SRIF may be found in a text by Gerald J.Bierman entitled "Factorization Methods for Discrete SequentialEstimation," which is Volume 28 of a series on Mathematics, Science andEngineering, published by Academic Press (1977). Some specificcomputational speed advantages accrue if one uses the SRIF approach, butthe equivalent least squares approach yields the same results and isused to describe the method here.

Some definitions are employed to enable compact equations to be used.The position vector, X, is a column vector whose components are therelative cartesian coordinates of the secondary antenna 30 with respectto the reference antenna 32. The column vector, H, is a measurementsensitivity vector whose components are a measure of how much themeasurement would change with a change in the respective x, y, and zcomponents of the position vector or matrix X.

Two column vectors E₁ and E₂ are defined to represent thefirst-difference phase measurements obtained from the first satelliteand the second satellite, respectively. In the ordinary use of leastsquares solutions, only one answer is obtained and E₁ and E₂ are singleelement vectors. However, in the example given above, it is known thatthere are a potential of 242 solutions. The specific whole-cyclemodified first-difference measurement from the first satellitecorresponding to each of these solutions must be an element of E₁, andsimilarly the whole-cycle modified first-difference measurements fromthe second satellite corresponding to each of the solutions must be anelement of E₂. In the case of the example above, 11 different values offirst-differences were found, but each of these values must be repeated22 times when forming E₁ so that each value can be paired with each ofthe 11 values of first-difference from the second satellite, and each ofthese needs to be paired with the mirror image. The same is of coursetrue when forming E₂.

For later use, standard rotation 3×3 rotation matrices are defined:R(a)_(z) is a coordinated rotation through and angle, a, correspondingto the azimuth of a satellite about the z axis; and R(e)_(x) is acoordinate rotation through an angle, e, corresponding to the complementof the elevation angle of the satellite about the x axis.

The 3×3 identity matrix, I, is defined as the all-zero matrix exceptthat the diagonal elements are one. The coordinate system chosen is inthe local tangent plane to the earth, with East, North and Upcorresponding to x, y and z cartesian coordinates, respectively.

For several reasons, it is advantageous to rotate the coordinate systemeach time a new satellite is processed. First, a rotation around the zaxis is performed so that the y axis points in the azimuth of thesatellite, and then a rotation around the x axis so that the z axispoints directly toward the satellite. This has the advantage that thesensitivity vector, H is sensitive to the rotated z component only, i.e.its x and y components are zero and the z component is one.

The normal sequential least squares equation is given by:

    A.sub.i X=V.sub.i                                          (1)

where the subscript i indicates which successive satellite'smeasurements are being incorporated. When no rotations are performedbetween successive satellites, A and V are defined by:

    A.sub.i =H.sub.i H.sub.i.sup.T +A.sub.i-1                  (2)

    V.sub.i =H.sub.i E.sub.i.sup.T +V.sub.i-1                  (3)

The superscript T means the matrix transpose and E_(i) means E₁ or E₂,depending on whether it is the first or second satellite, respectively.The initial values of the elements of the matrices A and V are allzeros.

When rotations are performed between the inclusion of measurements fromdifferent satellites, then equations (2) and (3) become:

    A.sub.i +HH.sup.T +R(e).sub.x R(a).sub.z A.sub.i-1 R(-a).sub.z R(-e).sub.x (4)

    V.sub.i =HE.sub.i.sup.T +R(e).sub.x R(a).sub.z V.sub.i-1   (5)

The subscript has been dropped from the H sensitivity vector, since itsdefinition is always the same in a coordinate system where the range tothe satellite only has a z component. The rotation angles defined by theazimuth and elevation angles are the azimuth and elevation angles asdefined in the coordinate system. Thus, they need to be adjusted for thecumulative rotations. The cosine and sine of the azimuth and elevationcan be computed directly from the direction of the satellites. As longas the direction cosines are rotated to each new coordinate system, thenthe azimuth and elevation angles will be correct. Use was made inequations (4) and (5) of the orthogonality constraint which causes thetranspose of a rotation matrix to be given the rotation matrix with thedirection of rotation reversed.

No matter how equation (1) is obtained, its solution is:

    X=A.sup.-a V                                               (6)

where A⁻¹ is defined as the inverse of A, such that

A⁻¹ A=I.

As stated above, orthogonality conditions require that the transpose ofthe rotation matrices is the same as the rotation around the negativeangle. Thus the matrix of solution vectors X can be converted back tothe original coordinate system at any point by rotating appropriately bythe negative of the azimuth of the last satellite included.

    X.sub.enu =R(-a).sub.z R(-e).sub.x X,                      (7)

where the subscript enu means in the east, north and up coordinates, andthe elevation and azimuth angles are defined in the east, north and upcoordinate system.

The incorporation of the first-difference measurements using equations(4) and (5) for the first satellite and then for the second satelliteresults in a least squares matrix equation which has all zero elementsin the x components. The reason is that measurements from only twosatellites are two-dimensional and correspond to the y,z plane.Obtaining the x component of the solutions involves a special process.Equation (6) is used to solve for the y and z components, which are thesecond and third rows of the matrix X. The first row corresponding tothe x components is zero.

The Pythagorean theorem is used to obtain the x components of thesolutions by constraining the solution to the appropriate sized sphere.A separation distance between the antennas of one meter, or 5.26wavelengths, results in a sphere of which the radius is the same, andgives the following equation: ##EQU1## If the value inside the squareroot is positive and

than some small threshold, then the positive square root is assigned toone of the pair of solutions with identical measurement components in E₁and E₂ and the negative square root is assigned to the other. If thevalue inside the square root is negative, and negative by more than somesmall threshold, then it dose not correspond to a valid solution and thepair of columns with the associated y and z components is removed fromthe X matrix and the V matrix so that their longitudinal dimension isreduced by two. If the value inside the square root is very near zero,with the specific threshold related to the noise in the carrier phasemeasurements, the value of the x component is set to zero in only one ofthe pair of columns of X with the associated y and z components. Thesecond column in the pair is removed from the potential solutions in thesame manner described above and the dimension of X and V is reduced byone. This last solution corresponds to the situation when the linedefined by the pair of first-difference equations does not pass throughthe sphere but just touches it at one point.

The elements in the top row and first column corresponding to the xcomponent of A are still zero and they cannot remain zero if thesolution is to be constrained to the spherical surface. Inserting asingle value in the element in the first column and the first row willconstrain the solution. A value of one in this element will make theconstraint equal in strength to the first-difference equations. Largervalues constrain the solution closer and closer to the sphere. Thechoice of a value of one is recommended since it allows the solution tomove slightly off the surface of the sphere to fit subsequentfirst-difference measurements from additional satellites as appropriateto noisy measurements from the first two satellites. Whatever the choiceof the first diagonal element of A, equation (1) must be reversed to setthe elements in the first row of B using the corresponding valuescomputed above for the elements of X.

    V=AX                                                       (9)

The complete set of potential equations using two satellites is nowdefined using the least squares equations. The matrices A, X and Vcontain all the required information in the form of equation (1) and canbe inverted for the specific coordinates in the form of equation (6).

The sequential processing of first-difference measurements fromadditional satellites can now be described. The process is very similarto the process given in equations (4) and (5) except that the processdescribed there must be modified to allow the appropriate constructionof the first-difference measurement vector E_(i) corresponding tosatellite i, where i sequentially takes on the values 3, 4, .... Firstthe incoming matrices A and V are rotated appropriately for the i'thsatellite.

    A.sub.i-1 =R(e).sub.x R(a).sub.z A.sub.i-1 R(-a).sub.z R(-e).sub.x(10 )

    V.sub.i-1 =R(e).sub.x R(a).sub.z V.sub.i-1                 (11)

A and V in equations (10) and (11) retain the same subscript since themeasurement data from the i'th satellite are still not included The onlychange caused by equations (10) and (11) is to rotate the coordinatesystem.

Now the rotated A and V matrices are used in equation (6) to giverotated coordinates of the potential solutions as the elements of X.Actually X could have been rotated directly, but since the rotated formsof A and V are needed later it is beneficial to rotate them and thensolve the equation (6). The solution (6) can be simplified if desired,since only the z components or third row elements of X are needed in thenext step.

Using the same example of a one-meter or 5.26 wavelengths separationdistance between antennas, it is clear that 11 different values of thefirst-difference measurement can be constructed. But a complete set ofpotential equations has already been found that spans the entirespherical surface. If subsequent satellites would result in differentsolutions, then they would agree with the solution from the first twosatellites. This would represent a false solution, since a true solutionmust agree with all of the measurements. Thus, each element of thefirst-difference measurement vector E_(i) must be selected from the 11different possible first-difference values, such that it is the valueclosest to the corresponding solution of in X. The motivation for thisrequirement is that any other value of the first difference shouldclearly disagree with the existing solution by at least one-half a cycleand would therefore, of necessity, be a false solution.

Since the first-difference values are measures of the distance of thesecondary antenna from the reference antenna in the direction of thesatellite, and since the z component is in the direction of thesatellite, the appropriate element of E_(i) is found by selecting fromthe ₁₁ possible values the one which is closest to the associated zcomponent of X. One could, if the closest possible value of thefirst-difference differs by more than a given amount, declare thesolution a false one, and remove it from further processing. However, itis safer to construct a complete E_(i) vector and incorporate it intothe potential solutions. This is done now that the E_(i) vector isconstructed by following equations (10) and (11) with:

    A.sub.i =HH.sup.T +A.sub.i-1                               (12)

    V.sub.i =HE.sub.i.sup.T +V.sub.i-1                         (13)

The resulting new values of A and V can be solved for updated values ofthe X, and the rms residuals of the solutions can be used to eliminatethose potential solutions which are highly unlikely to be the truesolution. An alternative is to compute the updated residual with respectto the last satellite only. This can be done by using equation (6) againto solve for new z components and differentiating them with thecorresponding value of E_(i). A positive threshold and its negativecould then be used to eliminate all potential solutions of which theresidual was greater than or less than the threshold. The thresholdvalue is a function of the measurement noise and is chosen such thatthere is a reasonable probability of eliminating false solutions and ahigh probability that the true solution is not eliminated.

After all the potential solutions are checked, the false solutionseliminated and the corresponding matrix and vector sizes reduced, thewhole process is repeated with the next satellite until thefirst-difference measurements from all the satellites have beenprocessed. Ideally, when all satellites have been processed only onesolution, the true solution, remains in the set of potential solutions.But even if multiple potential solutions remain, the true solution canstill be determined using subsequent measurements from the samesatellites.

Each of the potential solutions can be identified by three indices j, k,and l. The j index corresponds to the choice of whole-cycle value forthe first satellite's first-difference measurement, and thus in theexample would range between +5 and -5. The k index corresponds to thechoice of whole-cycle value for the second satellite and has the samerange as the j index. The l index has only two values, +1 and -1,corresponding to the choice of the plus square root or the minus squareroot in equation (8).

As long as the signals from each of the satellites are continuouslyreceived, the change in the whole cycles of the first differences isaccumulated in the measurements. This means that the indices from oneset of measurements can be related to the indices obtained at subsequenttimes from additional sets of measurements. But the changing geometry ofthe satellites relative to the antennas, due to either satellite motionor due to antenna motion, will cause the different false solutions toappear in the potential solution set. Only the true solution can remainin the potential solution set at each point in time. Thus the truesolution can be clearly identified as the geometry changes.

Once the true solution is identified, and as long as no loss of signalis encountered, the measurements can simply be modified by thewhole-cycle value indicated by the true solution indices, and themodified measurements fed into a standard Kalman filter or least squaressmoothing filter to obtain a continuing estimate of the orientation. Inan off-line modest rate the computations described above can beperformed to ensure that the signals have not been lost and that thetrue solution that is subsequently obtained has the same indices as thatbeing used in the active processing filters.

Kinematic Positioning System

The kinematic positioning system is only modestly different from theattitude determination system. The significant difference in theapparatus is that, in the kinematic system, the remote movable antennaand receiver are typically far enough removed from the reference antennaand receiver that they cannot be connected to the same reference clock.In addition, there is no separation distance constraint in the kinematicpositioning system, and the reference receiver location must remainfixed. The minimum number of satellites required to obtain a solutionwas two satellites for the attitude determination system. The kinematicpositioning system requires a minimum of four satellites. One satelliteis required for solution and removal of the differences in carrier phasecaused by the separate local reference clocks at the two receivers; andthree satellites are required to obtain the three-dimensionalcoordinates of the remote receiver antenna relative to the referencereceiver antenna. To allow a search through a set of potential solutionssimilar to that used in the attitude determination system, at least onemore satellite is required to provide a set of redundant measurements.

FIG. 6B is a block diagram showing how the method of the invention isused for kinetic position determination. In this case, the secondaryantenna 30 is movable with respect to the reference antenna 32, whichhas its own separate receiver 36' and processor 38'. The othersignificant difference is that, in addition to the four satellites 34shown, there is at least one pseudolite 35 installed at a fixed locationon the ground. Because the pseudolite 35 is much closer to the secondaryantenna 30 than the satellites, movement of the secondary antenna canprovide extremely rapid changes in the angular position of thepseudolite.

Before a set of potential solutions can be constructed for the kinematicpositioning system, an initial approximate position of the remotereceiver must be obtained. This was not required in the attitudedetermination system, because the distance between the antennas wasshort and held fixed at a specific value.

Several methods are available for establishing the initial approximateposition. If the remote receiver is moving, the best method isundoubtedly to use differential carrier smoother code measurements in astandard navigation Kalman filter implementation. If the remote receiveris stationary, then standard triple-difference or integrated Dopplerimplementation can yield competitive accuracy. Whatever method is usedto establish the approximate positions, it is important to alsodetermine the approximate uncertainty in the relative coordinate values.Typical strategies require that the uncertainty region over which thepotential solutions are evaluated have a range of at least plus or minusthree times the standard deviation in each dimension.

Once an initial position is established, four satellites are selected toestablish the initial set of potential solutions. The simplestimplementations usually involve the computations of thesecond-differences of the carrier measurements. First-differences of thecarrier phase measurements are computed just as they were in theattitude determination system for all satellites. Specifically, thecarrier phase measurement at the reference site for each satellite issubtracted from the carrier phase measurement at the remote site for thecorresponding satellite. Then, in order to remove the effects ofdifferent reference clock phase at the two receivers, thefirst-difference of carrier phase measurements from one of thesatellites selected as a reference satellite is subtracted from thefirst-difference carrier phase measurements from each of the othersatellites. The result is a set of three second-difference carrier phasemeasurements.

Just as the first-difference carrier phase measurements were ambiguousover a region of two-dimensional space in the attitude determinationsystem, so the second-difference phase measurements are ambiguous in aregion of three-dimensional space. For illustrative purposes, assumethat the uncertainty region is a three-dimensional volume about twometers in length in each direction. Assume also that only L₁ signals areobtained by the receivers and that the carrier phase wavelength isapproximately 19 centimeters. The second-difference carrier phasemeasurements associated with each of the satellite pairs could then haveabout 11 acceptable whole-cycle values that would intersect theuncertainty region. Thus, the number of potential solutions would be 11³or 1,331 potential solutions.

No special process is required to obtain each of these solutions, as wasthe case in the attitude determination system. The measurement vectorsE₁, E₂ and E₃ are each 1,331 elements long and are formed by choosingall possible difference measurements. The measurements associated withthe first satellite pair of second-difference measurements are enteredas elements of E₁, and the second satellite pair with E₂, and the thirdsatellite pair with E₃. The coordinates are rotated and the measurementsare incorporated into A and V via equations (4) and (5), just as theywere in the attitude determination system. After the measurements fromthe four satellites are incorporated via the three sets ofsecond-difference measurements, the least square equation for the matrixX could be solved using equation (6) for the set of coordinates of allthe potential solutions, though the solutions is not explicitly requiredat this point.

The next step is to commence the elimination of false solutions, inalmost exactly the same way as in the attitude determinationapplication. However, instead of using first-differences of theredundant satellites, the second-differences are formed by subtractingfrom each of the redundant first-differences the first-differencesmeasurement obtained from the same reference satellite that was usedearlier in forming the original second-difference measurements. This newredundant set of second-difference measurements is ambiguous andpresumably can have, for example, 11 different possible whole-cyclevalues. As described previously for the attitude determination system,the A and V matrices are rotated using equations (10) and (11). Then therotated least squares matrix is solved for X to obtain the z componentof each potential solution. The second-difference measurement vectorE_(i) is constructed by choosing for each element the specificwhole-cycle value of the second-difference measurement closest to theassociated z component in the X matrix.

Equations (12) and (13) are then implemented and, based on theresiduals, computed in the same fashion as in the attitude determinationsolution, false values being detected and eliminated. The above processis repeated for each of the redundant satellites and, ideally, only thetrue solution will remain after all of the redundant satellites havebeen processed.

Just as in the attitude determination system, if there are multiplepotential solutions left when all of the redundant satellites have beenprocessed it is still possible to identify the correct solution if oneis willing to wait while the satellite geometry changes. Repeating theentire process using data collected at subsequent time intervals leadsto new sets of potential solutions and indices, as described earlier,can be constructed relating the potential solutions between data sets.Only the true solution will continue to remain a common element in thesets of the potential solutions.

Once the true solution is identified, the modified second-differenceequations are sent directly to a Kalman filter or least squaressmoothing filter for a continuing smoother output of relative remotereceiver coordinates. The process described above can continue to becomputed at a slow rate to ensure that an inadvertent loss of signal hasoccurred. Such a loss of signal would be signaled by a new true solutionwhose indices would be different from the prior true solution.

Summary of Ambiguity Resolution

The method steps for resolving whole-cycle ambiguity in kinematicpositioning and, incidentally, in attitude determination, are shown inflowchart form in FIG. 7. Basically, these are the steps performed inthe processor 38 (FIG. 6B).

The first step in the processing loop illustrated in FIG. 7 is, as shownin block 50, to measure the code an carrier phase for L₁ (and L₂ ifavailable) from each satellite. In decision block 52, the question posedis whether initial ambiguities is the relative position of the secondaryantenna with respect to the reference antenna have been resolved.Clearly, on a first pass through this flowchart the answer will be inthe negative. If the answer is affirmative, the relative positioninformation is passed to a standard Kalman filter for smoothing andoutput, as indicated in block 54. If ambiguities have not yet beenresolved, another decision block 56 poses the question whether this isan attitude determination system or a kinematic positioning system.Normally, these systems would not be incorporated into one processor,but the flowchart is intended to show the commonality of the twosystems. In an attitude determination system, only two satellites areinitially employed, as indicated in block 58, to compute a set ofpotential solutions, located in an uncertainty region on the surface ofa sphere. In a kinematic positioning system, the next step is to computean approximate relative position for the remote antenna, as indicated inblock 60; then to use four satellites to compute a set of potentialsolutions located in a three-dimensional uncertainty volume, asindicated in block 62.

The remaining blocks in the flowchart are common to both types ofsystems. The first of these blocks, at 64, performs a coordinaterotation such the z (vertical) axis is pointed directly to a selectedredundant satellite, and for each measurement the processor solves forthe z component. Then, in block 66, the processor constructs ameasurement vector by choosing a whole-cycle value nearest to the zcomponent for each potential solution. The resulting new measurement isincorporated into the matrix of potential solutions, as in block 68.Then, false solutions are edited out of the matrix using a residualthreshold, as indicated in block 70. In block 72, a check is made todetermine if additional redundant measurements remain to be processedfor the redundant satellite under consideration. If so, return is madeto block 64 to continue processing. If not, a check is made in block 74to determine if only one potential solution remains. If so, theambiguity has been resolved and this part of the processing is complete,as indicated at 76, and processing is continued in block 54, using thestandard Kalman filter. If more than one solution still remains,processing continues in block 78, where a comparison is made forintersection of prior solutions, i.e. solutions reached at earlier timeintervals and different satellite geometries. Implied in this block isthe further elimination false solutions based on their failure to recurat successive time intervals. In decision block 80, another check ismade for the existence of a single solution. If the ambiguity has beentotally resolved, processing continues via blocks 76 and 54. If there isstill ambiguity, processing continues in block 50, using data from adifferent redundant satellite.

Because the geometry of the antenna system changes so rapidly withrespect to the pseudolite 35, false solutions can be eliminated muchmore rapidly than if four real satellites are used. The invention worksbest when the pseudolite or pseudolites are used in addition to foursatellites. The satellites provide a set of solutions and the additionalone or more pseudolites provide redundancy in two respects: first byproviding an additional one or more set of measurements, equivalent tohaving one or more additional satellites, and secondly by providingstill further sets of different measurements as the secondary antenna ismoved to new positions and the angular positions of the pseudoliteschanges. In the case of a secondary antenna located on a movingaircraft, the use of one or more pseudolites reduces the ambiguityresolution time to a matter of seconds, as compared with minutes whenthe pseudolites are not used. To be effective, the pseudolites must bepositioned to provide a rapid angular position change as the secondaryantenna moves along its normally intended path. For example, apseudolite should not be located in line with an aircraft's finalapproach, or the angular position might not change at all during acritical phase of the landing. But if a pseudolite is positioned well toone side of the runway, the rate of change of angular position can bemaximized for a critical point in the approach path. Multiplepseudolites can be appropriately positioned for landing on multiplerunways.

It will be appreciated from the foregoing that the present inventionrepresents a significant advance in the field of satellite-baseddifferential positioning systems. In particular, the invention providesa novel technique for removing whole-cycle ambiguity from carrier phasemeasurements of relative position, in kinematic positioning systems, byusing one or pseudolites to reduce the time needed to resolve theambiguity. It will also be appreciated that, although an embodiment ofthe invention has been described in detail for purposes of illustration,various modifications may be made without departing from the spirit andscope of the invention. Accordingly, the invention is not to be limitedexcept as by the appended claims.

I claim:
 1. A method for determining the relative position of asecondary receiving antenna with respect to a reference receivingantenna in a satellite-based positioning system, the method comprisingthe steps of:making carrier phase measurements based on the reception ofa carrier signal from each of a plurality N of satellites, where N isthe minimum number of satellites needed to compute the relative positionof the secondary antenna; deriving from the carrier phase measurementsan initial set of potential solutions for the relative position, whereinthe initial set of potential solutions all fall within a region ofuncertainty defined by a sphere having a radius equal to the maximumdistance between the two antennas, and wherein multiple potentialsolutions arise because of whole-cycle ambiguity of the carrier signal;positioning at least one pseudolite transmitter on the ground at alocation near an intended path of travel of the secondary antenna;making redundant carrier phase measurements based on the reception of acarrier signal from the at least one pseudolite; and eliminating falsesolutions from the initial set of potential solutions, based on acomparison of the redundant carrier phase measurements with the initialset of potential solutions, to reduce number of potential solutions toclose to one, whereby the number of potential solutions is not increasedby use of the redundant carrier phase measurements.
 2. A method asdefined in claim 1, and further comprising the steps of:making redundantcarrier phase measurements based on the reception of a carrier signalfrom other additional satellites; and eliminating other false solutionsfrom the set of potential solutions, based on a comparison of theadditional redundant carrier phase measurements with the initial set ofpotential solutions.
 3. A method as defined in claim 1, and furthercomprising the steps of:comparing items in the set of potentialsolutions, including those obtained from the redundant phasemeasurements, with solutions obtained in a prior time interval, toprovide another basis for eliminating false solutions; wherein thelocation of the at least one pseudolite provides for a rapidly changingangular geometry and reduces the time needed to eliminate falsesolutions.
 4. A method as defined in claim 1, wherein:the step ofderiving an initial set of potential solutions includes locating pointsof intersection of planar carrier wavefronts defining possible locationsof the secondary antenna within the region of uncertainty; and the stepof eliminating false solutions includes, locating a set of planarcarrier wavefronts from the at least one pseudolite such that thewavefronts also define possible locations of the secondary antennawithin the region of uncertainty, selecting, for each of the initial setof potential solutions, a planar carrier wavefront from the additionalsatellite such that the selected wavefront is the one closest to thepotential solution, and disregarding each potential solution for whichthe closest wavefront from the additional satellite is spaced from thepotential solution by more than a selected threshold.
 5. A method asdefined in claim 1, wherein:the initial set of potential solutions isinitially stored in a local tangent coordinate system, x, y, z, where zis a vertical axis; and the method further comprises, after the step ofmaking redundant carrier phase measurements, the additional step ofrotating the coordinate system of the set of potential solutions, topoint the z axis toward the additional satellite and thereby facilitatethe step of eliminating false solutions.
 6. A method as defined in claim4, wherein:the initial set of potential solutions is initially stored ina local tangent coordinate system, x, y, z, where z is a vertical axis;and the method further comprises, after the step of making redundantcarrier phase measurements, the additional step of rotating thecoordinate system of the set of potential solutions, to point the z axistoward the additional satellite and thereby facilitate the step ofeliminating false solutions.
 7. A method as defined in claim 6, whereinthe step of selecting a planar carrier wavefront from the additionalsatellite is performed by comparing the z-axis component of eachpotential solution with possible positions of wavefronts in the regionof uncertainty.
 8. A method as defined in claim 1, wherein:the methodfurther comprises the initial step of determining the approximateinitial relative position of the secondary antenna.
 9. A method asdefined in claim 8, wherein:the step of determining the approximateposition includes making measurements of pseudorandom codes receivedfrom the satellites, and computing the approximate position from thecode measurements.
 10. A method as defined in claim 1, and furthercomprising the steps of:positioning the reference receiving antenna andan associated reference receiver close to the pseudolite; and using thepseudolite signal to transmit measurement and correction data from thereference receiver to the secondary antenna.